Robust containment control in a leader–follower network of...

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2016; 26:3791–3805Published online 7 March 2016 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.3535

Robust containment control in a leader–follower network ofuncertain Euler–Lagrange systems

Justin R. Klotz*,†, Teng-Hu Cheng and Warren E. Dixon

Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, USA

SUMMARY

A distributed controller is developed that yields cooperative containment control of a network of autonomousdynamical systems. The networked agents are modeled with uncertain nonlinear Euler–Lagrange dynamicsaffected by an unknown time-varying exogenous disturbance. The developed continuous controller is robustto input disturbances and uncertain dynamics such that asymptotic convergence of the follower agents’ statesto the dynamic convex hull formed by the leaders’ time-varying states is achieved. Simulation results areprovided to demonstrate the effectiveness of the developed controller. Copyright © 2016 John Wiley &Sons, Ltd.

Received 18 September 2014; Revised 9 November 2015; Accepted 31 January 2016

KEY WORDS: nonlinear control; distributed control; robust control; containment control

1. INTRODUCTION

Cooperation of autonomous systems has been widely investigated in recent years because of itsapplicability to many engineering applications. Examples include the distributed control of infras-tructure systems, industrial process control, vehicle formation control, and social network analysis.Instead of a single control system performing a task, multiple potentially lower-cost systems can becoordinated to achieve a network-wide goal, such as employing a team of unmanned aerial vehiclesor autonomous underwater vehicles (AUVs) to perform surveillance, search and rescue, or haz-ardous material recollection. Consensus is a network control strategy introduced to quantify sucha cooperative objective where the states of the networked systems are regulated to the same valuewhile network neighbors simultaneously tend to each others’ states during transient performance. Inconsensus, network interaction is commonly modeled as being distributed, where systems only useinformation from network neighbors to compute a control signal such that a global objective can beachieved without requiring all-to-all communication; see [1–4] for some prominent examples.

Although terminology has been inconsistent in the literature, synchronization (cf. [5–8]) typicallyrefers to the generalization of the consensus problem by allowing the desired state of the networkedsystems to be time-varying. The desired trajectory for the cooperating systems is typically specifiedby a network leader, which can be a preset time-varying function or a physical system that the‘follower’ systems interact with via sensing or communication. For example, a task that requiresan expensive sensor in a search and rescue mission can be accomplished by endowing just onesystem with the expensive sensor and instructing the other systems to cooperatively interact withthe autonomous ‘leader.’ This control objective is made more practical by limiting interaction withthe leader to only a subset of the follower systems.

*Correspondence to: Justin R. Klotz, Department of Mechanical and Aerospace Engineering, University of Florida,Gainesville, FL 32611-6250, USA.

†E-mail: [email protected]

Copyright © 2016 John Wiley & Sons, Ltd.

3792 J. R. KLOTZ, T. CHENG, AND W. E. DIXON

Containment control (cf. [9–18]) is a generalization of the synchronization problem that allowsfor a collection of leaders. Containment control is useful in applications where a team of autonomousvehicles is directed by multiple pilots or for networks of autonomous systems where only a subset ofthe systems is equipped with expensive sensing hardware. The typical objective in the containmentcontrol framework is to regulate the states of the networked systems to the convex hull spanned bythe leaders’ states, where the convex hull is used because it facilitates a convenient description ofthe demarcation of where the follower systems should be with respect to the leaders.

Containment control is investigated in [9] and [11] for static leaders and in [10] for a combi-nation of static and dynamic leaders. Containment controllers for dynamic leaders and followerswith linear dynamics are developed in [12–15, 19]. A controller designed for the containment ofsocial networks with linear opinion dynamics represented with fractional order calculus is devel-oped in [16]. Results in [17] and [18, 20] develop a model knowledge-dependent and model-freecontainment controller, respectively, for the case of dynamic leaders and Euler–Lagrange dynamics,where Euler–Lagrange dynamics are used for the broad applicability to many engineering systems.However, none of the previous results analyze the case where follower systems are affected by anexogenous, unknown disturbance, which has the capability of cascading and disrupting the perfor-mance of the entire network from a single source. Compared with the work in [18], the developmentin this work demonstrates compensation of unknown input disturbances and does not require com-munication of an acceleration signal from the leader or neighboring follower agents. Furthermore,whereas the stability analysis in [18] is temporally divided into an estimation segment and a sub-sequent Lyapunov-based stability analysis for showing network containment, which relies on theassumption of boundedness of the dynamics until estimate equivalence is reached, the present workyields asymptotic network containment throughout the entire state trajectory. A sliding-mode basedcontroller is developed in [20] for the compensation of exogenous disturbances in containment con-trol of an undirected network of agents with Euler–Lagrange dynamics. However, the sliding-modeportion of the controller in [20] makes the developed control law discontinuous, which is diffi-cult to implement in practice. Compared with the development in [20], the controller developed inthis paper provides compensation of unknown exogenous disturbances with a continuous controlsignal, which is generally much more feasible to implement. The contribution of this paper is thedevelopment of a continuous, distributed controller that provides asymptotic containment controlin a network of dynamic leaders and followers with uncertain nonlinear Euler–Lagrange dynamics,despite the effects of exogenous disturbances, where at least one of the followers interacts with atleast one leader and the follower network is connected. Euler–Lagrange equations of motion areused to model the agents’ dynamics because of their ability to capture nonlinearities intrinsic tomany engineering systems, such as robotic systems and power generators (cf. [21]), and are still thesubject of active research efforts (cf. [22–24]).

2. PROBLEM FORMULATION

2.1. Preliminaries

Graph theory is used to streamline the analysis of the considered networked dynamical systems.To facilitate the subsequent analysis, consider a network of L 2 Z>0 leader agents and F 2 Z>0follower agents. Communication of the follower agents is described with a fixed undirected graph,GF D ¹VF ; EF º, where VF , ¹LC 1; : : : ; LC F º is the set of follower nodes and EF � VF �VFis the corresponding edge set. An undirected edge .i; j / (and also .j; i/) is an element of EF ifagents i; j 2 VF communicate information with each other; without loss of generality, the graphis considered to be simple, that is, .i; i/ … EF 8i 2 VF . The follower agent neighbor set NF i ,¹j 2 VF j .j; i/ 2 EF º is the set of follower agents that transmit information to agent i . The connec-tions in GF are succinctly described with the adjacency matrix AF D

�aij�2 RF�F , where aij > 0

if .j; i/ 2 EF and aij D 0 otherwise. The Laplacian matrix LF D�pij�2 RF�F associated

with graph GF is constructed such that pi i DPj2NFi aij and pij D �aij if i ¤ j . The directed

graph G D ¹VF [ VL; EF [ ELº containing both the leader and follower agents is a supergraph of

Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2016; 26:3791–3805DOI: 10.1002/rnc

ROBUST CONTAINMENT CONTROL IN LEADER–FOLLOWER NETWORKS 3793

GF constructed by appending an edge .l; i/ 2 EL to GF if leader agent l 2 VL communicates infor-mation to follower agent i 2 VF , where VL , ¹1; : : : ; Lº is the leader node set and EL � VL � VFis the set of leader–follower edges. The adjacency matrix A D

�aij�2 R.LCF /�.LCF / for graph

G is similarly defined such that aij > 0 if .j; i/ 2 EF [ EL and aij D 0 otherwise. Let the diag-onal leader-connectivity matrix B D

�bij�2 RF�F be defined such that bi i D

Pl2VL ail and

bij D 0 for i ¤ j . The Laplacian matrix for graph G can be constructed similarly to LF and can

be represented as L D�

0L�L 0L�FLL LF C B

�, where LL 2 RF�L and 0 is the zero matrix of denoted

dimensions. For brevity, let the matrix LF C B be abbreviated as LB , LF C B .

2.2. Dynamic model and properties

The dynamics of each follower agent i 2 VF are described by the nonidentical Euler–Lagrangeequations of motion

Mi .qi / Rqi C Ci .qi ; Pqi / Pqi CHi . Pqi /CGi .qi /C di D �i ; (1)

where qi 2 Rm is the generalized configuration coordinate, Mi W Rm ! Rm�m is the inertiamatrix, Ci W Rm � Rm ! Rm�m is the Coriolis/centrifugal matrix, Hi W Rm ! Rm representsfriction, Gi W Rm ! Rm represents gravitational effects, �i 2 Rm represents the vector of controlinputs, and di W R>0 ! Rm is a time-varying nonlinear exogenous disturbance. Functional depen-dency will be omitted in the remainder of the paper where the meaning is clear from context.The following assumption is characteristic of physical systems with dynamics described by Euler–Lagrange equations of motion and is used to provide bounds on the effects of the inertia matrix ofthe dynamics of each system (cf. [11, 18, 25]).

Assumption 1For each follower agent i 2 VF , the inertia matrix Mi is symmetric, positive-definite, and satisfiesthe inequalitymi k�k

2 6 �TMi� 6 Qmi k�k2 8� 2 Rm, where k�k represents the standard Euclideannorm, mi 2 R>0 is a known constant, and Qmi W Rm ! R>0 is a bounded function such thatmi 6 Qmi .qi / 6 Nmi 8qi 2 Rm, where Nmi 2 R>0 is a known constant.

The following assumptions concern the smoothness of the agents’ dynamics. For example,marine craft hydrodynamics and exogenous disturbances (e.g., waves) are often modeled as smooth(cf. [26]).

Assumption 2 ([25])For each follower agent i 2 VF , the functions Mi ; Ci ;Hi ; Gi are second-order differentiable suchthat the second time derivative is bounded, provided q.k/i 2 L1, k D 0; : : : ; 3.

Assumption 3 ([27])For each follower agent i 2 VF , the time-varying disturbance term is sufficiently smooth such thatit and its first two time derivatives, di ; Pdi ; Rdi , are bounded by known constants.

The following assumption specifies the smoothness and boundedness of the leader trajectories.For example, in a UAV surveillance mission, the monitored area is naturally bounded.

Assumption 4Each time-varying leader configuration coordinate, ql W R>0 ! Rm .l 2 VL/, is sufficiently smoothsuch that ql 2 C2; additionally, each leader configuration coordinate and its first two time derivativesare bounded such that ql ; Pql ; Rql 2 L1.

The following assumption is necessary for the leaders’ states to be able to affect the trajectoriesof the follower agents.

Assumption 5For each follower agent i 2 VF , there exists a directed path from a leader l 2 VL to i .



Note that by Assumptions 5 and [9, Lemma 4.1], the matrix LB is positive-definite. Forconvenience, the follower agents’ dynamics are stacked as

M RQF C C PQF CH CG C d D �; (2)

where

M ,diag ¹MLC1; : : : ;MLCF º 2 RFm�Fm

QF ,�qTLC1; : : : ; q

TLCF

�T2 RFm

C ,diag ¹CLC1; : : : ; CLCF º 2 RFm�Fm

H ,�HTLC1; : : : ; H

TLCF

�T2 RFm

G ,�GTLC1; : : : ; G

TLCF

�T2 RFm

d ,�dTLC1; : : : ; d

TLCF

�T2 RFm

� ,��TLC1; : : : ; �

TLCF

�T2 RFm:

3. CONTROL OBJECTIVE

The objective is to design a continuous controller for the follower agent dynamics in (1) that drivesthe states of all follower agents to the (possibly time-varying) convex hull spanned by the leaderagents’ states despite exogenous nonlinear input disturbances and modeling uncertainties. Further-more, only the configuration coordinate and its first time derivative are assumed to be measurablefor the leader and follower agents. An error signal, e1;i 2 Rm .i 2 VF /, is developed to quantifythe neighborhood tracking error as

e1;i ,X

j2VF[VL

aij�qi � qj

�; (3)

which includes the state difference between neighboring follower agents and neighboring leaderagents. Note that there is no restriction on an edge weight aij for an existing connection.i; j / 2 E other than the weight is positive and aij D aj i 8i; j 2 VF . Therefore, the controlcan emphasize a connection .j; i/ by increasing aij if it is desired for agent i 2 VF to maintainclose similarity to agent j 2 VF [ VL . An auxiliary tracking error, e2;i 2 Rm, is designed as

e2;i , Pe1;i C ˛1;ie1;i ;

where ˛1;i 2 R>0 is a constant gain. By stacking the follower agents’ error signals e1;i and e2;i

as E1 ,heT1;LC1; : : : ; e

T1;LCF

iT2 RFm and E2 ,

heT2;LC1; : : : ; e

T2;LCF

iT2 RFm, the network

error dynamics can be written as

E1 D .LB ˝ Im/QF C .LL ˝ Im/QL; (4)

E2 D PE1 C�1E1; (5)

where ˝ represents the Kronecker product, I is the identity matrix of denoted dimensions, QL ,�qT1 ; : : : ; q

TL

�T2 RLm is the stack of leader agent states, and�1 , diag ¹˛1;LC1; : : : ; ˛1;LCF º˝

Im 2 RFm�Fm is a diagonal matrix of gains. An auxiliary error signal, R 2 RFm, is designed as

R , .LB ˝ Im/�1�PE2 C�2E2

�; (6)



where the matrix LB ˝ Im is invertible because LB is invertible and �2 ,diag ¹˛2;LC1; : : : ; ˛2;LCF º˝Im 2 RFm�Fm is a diagonal matrix containing the gains ˛2;i 2 R>0.The auxiliary error signal R is not used in the subsequently designed controller because it dependson the second derivative of the configuration coordinate and is only introduced to facilitate anexpression for the closed-loop error system.

Similar to [15], the error system in (3) is designed such that kE1k ! 0 implies that the Euclideandistance from qi to the convex hull formed by the leader agents also asymptotically convergesto zero for all i 2 VF . This implication is stated in the following lemma, which is used in thesubsequent development.

Lemma 1If Assumption 5 is satisfied, then kE1k ! 0 implies that d .qi ;Conv ¹ql j l 2 VLº/! 0 8i 2 VF ,where Conv¹�º denotes the convex hull of the set of points in its argument and the distance d .p; S/between a point p and a set S is defined as infs2S kp � sk for all p 2 Rn and S � Rn.

ProofSee the Appendix. �

4. CONTROLLER DEVELOPMENT

An open-loop error system is designed by pre-multiplying the auxiliary tracking error R in (6) byM and using (2), (4), and (5) as

MR D � � d C S1 C S2; (7)

where the functions S1 2 RFm and S2 2 RFm are defined as

S1 ,M .QF / .LB ˝ Im/�1�.�1 C�2/E2 ��

21E1

�� C

�QF ; PQF

�PQF �H

�PQF

��G .QF /

CM .QF /�L�1B LL ˝ Im

�RQL CH

��L�1B LL

�˝ Im

�PQL

�CG

��L�1B LL

�˝ Im

�QL

�C C

��L�1B LL

�˝ Im

�QL;

��L�1B LL

�˝ Im

�PQL

� ��L�1B LL

�˝ Im

�PQL

�M��L�1B LL

�˝ Im

�QL

� �L�1B LL ˝ Im

�RQL;

S2 , � C��L�1B LL

�˝ Im

�QL;

��L�1B LL

�˝ Im

�PQL

��L�1B LL

�˝ Im

�PQL �H

��L�1B LL

�˝ Im

�PQL

��G

��L�1B LL

�˝ Im

�QL

�CM

��L�1B LL

�˝ Im

�QL

� �L�1B LL ˝ Im

�RQL:

Terms in (7) are organized so that, after a mean value theorem-based approach (cf. [28, Lemma 5]),kS1k can be upper-bounded by a function of the errors signals E1; E2; R and kS2k can beupper-bounded by a constant. Note that the term M .QF / .LB ˝ Im/

�1�.�1 C�2/E2 ��

21E1

�can be upper-bounded by a function of the error signals using a mean value theorem-based approach after adding and subtracting the term M

��L�1B LL

�˝ Im

�QL

�.LB ˝ Im/

�1

��.�1 C�2/E2 ��

21E1

�within S1.

The developed robust distributed controller for follower agent i 2 VF is designed as

�i DXj2NFi

aij��ks;j C Im

�e2;j � .ks;i C Im/ e2;i

�� bi i .ks;i C Im/ e2;i C �i ; (8)

where the function‡ �i W ˘jNFi jC1jD1 Rm ! Rm is the generalized solution to the differential equation

‡In this context,˘ denotes the Cartesian product.



P�i ,Xj2NFi

aij��ks;j C Im

�˛2;j e2;j � .ks;i C Im/ ˛2;ie2;i

�C

Xj2NFi

aij��jSgn

�e2;j

�� iSgn .e2;i /

�� .ks;i C Im/ bi i˛2;ie2;i � bi i�iSgn .e2;i /

(9)

with �i .0/ D �i0 2 Rm as a user-specified initial condition, where ks;i 2 Rm�m is a con-stant positive-definite gain matrix, �i 2 Rm�m is a constant diagonal positive-definite gainmatrix, the function Sgn.�/ is defined for all � D

��1; : : : ; �v

�T2 Rv as Sgn.�/ ,�

sgn .�1/ ; : : : ; sgn .�v/�T

, and j � j denotes set cardinality for a set argument. Note that the con-troller in (8) is continuous, only relies on the configuration coordinate and its first derivative, and isdistributed in communication: agent i requires its own error signal and the error signals of neighborsj 2 NF i . The use of neighbors’ error signals in the control law provides cooperation among the fol-lower agents. Assuming that a neighbor’s state can be sensed, then only one-hop communication isnecessary to compute the control authority in (8). In (8) and (9), the terms multiplied by the gain ks;iprovide proportional and derivative feedback and the signum-based terms multiplied by the gain �iprovide robust feedback that is used to reject the unknown time-varying disturbances, as shown inthe following stability analysis. Similar to how sliding-mode-based controllers use the signum func-tion to compensate for exogenous disturbances, the developed approach uses the signum functionin the derivative of the controller to compensate for exogenous disturbances, provided that they aresufficiently smooth (Assumption 3). Note that a strategy involving additive gradient-based controlterms, such as that in [29], can be used if collision avoidance is necessary for the control objective.

After taking the time-derivative of (7), the closed-loop error system is

M PR D� .LB ˝ Im/ .Ks C IFm/�PE2 C�2E2

�� .LB ˝ Im/ ˇSgn .E2/C QN C .LB ˝ Im/Nd

� .LB ˝ Im/E2 �1

2PMR;

(10)

where the first two terms are contributions from the derivative of the stack of followeragents’ control inputs P� D .LB ˝ Im/ .Ks C IFm/

�PE2 C�2E2

�� .LB ˝ Im/ ˇSgn .E2/;

Ks , diag .ks;LC1; : : : ; ks;LCF / 2 RFm�Fm is a block-diagonal gain matrix and ˇ ,diag .�LC1; : : : ; �LCF / 2 RFm�Fm is a diagonal gain matrix; and the unknown auxiliary functionsQN W ˘7

jD1RFm ! RFm and Nd W R>0 ! RFm are defined as

QN , PS1 C .LB ˝ Im/E2 �1

2PMR; (11)

Nd ,�L�1B ˝ Im

� �Pd C PS2

: (12)

Terms in QN are segregated such that after taking advantage of the expressions QF D�L�1B ˝ Im

�E1�

�L�1B ˝ Im

�.LL ˝ Im/QL, PE1 D E2��1E1, and PE2 D .LB ˝ Im/R��2E2,

Assumptions 2 and 4, and a mean value theorem-based approach (cf. [28, Lemma 5]), (11) can beupper-bounded by QN 6 � .kZk/ kZk ; (13)

where Z 2 R3Fm is the composite error vector defined as Z ,�ET1 ET2 RT

�Tand � W R>0 !

R>0 is a strictly increasing, radially unbounded function. Moreover, other terms in (10) are segre-gated in the function Nd such that it and its first derivative can be upper-bounded such that, for allk 2 ¹1; : : : ; Fmº,

supt2Œ0;1/

jNd jk 6 ıa;k;



supt2Œ0;1/

ˇPNdˇk6 ıb;k;

after using Assumptions 3 and 4, where j � jk denotes the absolute value of the kth componentof the vector argument and ıa;k; ıb;k 2 R>0 are constant bounds. In the subsequent stabil-ity analysis, the terms in Nd and PNd are compensated by using the signum feedback terms in(9). For clarity in the following section, let the vectors �a;i ; �b;i 2 Rm be defined such that�a;i ,

�ıa;m.i�1/C1 : : : ; ıa;m.i�1/Cm

�Tand �b;i ,

�ıb;m.i�1/C1 : : : ; ıb;m.i�1/Cm

�T, which

represent the contribution of the disturbance terms for each agent.

5. STABILITY ANALYSIS

An auxiliary function P W RFm � RFm � R>0 ! R is included in the subsequently definedcandidate Lyapunov function so that sufficient gain conditions may be obtained for the compensa-tion of the bounded disturbance terms in Nd . Let P be defined as the generalized solution to thedifferential equation

PP D ��PE2 C�2E2

�T.Nd � ˇSgn .E2// ;

P.0/ D

FmXkD1

ˇk;k jE2.0/jk �ET2 .0/Nd .0/;

(14)

where ˇk;k denotes the kth diagonal entry of the diagonal gain matrix ˇ. Provided the sufficient gaincondition in (18) is satisfied, then P > 0 for all t 2 Œ0;1/ [8].

Remark 1Because the closed-loop error system in (10) and the derivative of the signal P in (14) are dis-continuous, the existence of Filippov solutions in the given differential equations is addressedbefore the Lyapunov-based stability analysis is presented. Consider the composite vector ,�ZT ; �TLC1; : : : ; �

TLCF ;

pP�T2 R4FmC1, composed of the stacked error signals, the signal

contributing discontinuities to the derivative of the developed controller, and the aforementionedauxiliary signal P . Existence of Filippov solutions for the closed-loop dynamical system P DK Œh1 .; t/ can be established, where h1 W R4FmC1 � R>0 ! R4FmC1 is a function defined asthe right-hand side of P and K Œh1 .%; t/ , \ı>0 \�.Sm/D0 co h1 .Bı.%/ n Sm; t /, where ı 2 R,\�.Sm/D0 denotes the intersection over the sets Sm of Lebesgue measure zero, co denotes convexclosure, and Bı.%/ ,

®� 2 R4FmC1 j k% � �k < ı

¯, where �; % 2 R4FmC1 are used as dummy

variables [30–32].

Let the auxiliary gain constant ˆ 2 R be defined as

ˆ , min

²mini2VF

˛1;i �1

2; mini2VF

˛2;i �1

2; �min

�L2B

�³;

where �min.�/ denotes the minimum eigenvalue. A continuously differentiable, positive-definitecandidate Lyapunov function VL W D! R is defined as

VL .y; t/ ,1

2ET1 E1 C

1

2ET2 E2 C

1

2RTM.t/RC P; (15)

where the composite vector y 2 R3FmC1 is defined as y ,�ZTpP�T

, D is defined as the openand connected set

D ,°� 2 R3FmC1 j k�k < inf

��1

°�h2pˆ�min ..LB ˝ Im/Ks .LB ˝ Im//

;1

±±;



and ��1¹�º denotes the inverse mapping of a set argument. To facilitate the description of the semi-global property of the following Lyapunov-based stability analysis, the set of stabilizing initialconditions SD � D is defined as

SD ,²� 2 D j k�k <

s�1

�2inf��1

°�h2pˆ�min ..LB ˝ Im/Ks .LB ˝ Im//

;1

±³:

Because of the construction of VL in (15), VL satisfies the inequalities

�1 kyk2 6 VL .y; t/ 6 �2 kyk2 8t 2 Œ0;1/ ; (16)

where �1; �2 2 R>0 are constants defined as �1 , 12

min®1;minj2VF mj

¯and �2 ,

max®1; 12

maxj2VF Nmj¯

via Assumption 1. The following theorem describes the performance ofthe networked dynamical systems through the use of the Lyapunov function candidate in (15).

Theorem 1For every follower agent i 2 VF , the distributed controller in (8) guarantees that all signals arebounded under closed-loop control and that containment control is semi-globally achieved in thesense that d .qi ;Conv ¹ql j l 2 VLº/! 0 as t !1, provided that the gains ks;i , for all i 2 VF , areselected sufficiently large such that the initial condition y.0/ lies within the set of stabilizing initialconditions SD, and the gains ˛1;i , ˛2;i , �i are selected according to the sufficient conditions

˛1;i >1

2; ˛2;i >

1

2; (17)

�min .�i / > k�a;ik1 C1

˛2;i

�b;i1 (18)

for all i 2 VF , where k�k1 denotes the infinity norm.

ProofUsing Filippov’s framework, a Filippov solution can be established for the closed-loop system Py Dh2 .y; t/, where h2 W R3FmC1�R>0 ! R3FmC1 denotes the right-hand side of the derivative of theclosed-loop error signals and PP . Accordingly, the time derivative of (15) exists almost everywhere(a:e:) on the time domain Œ0;1/ and PVL

a:e:2 PQVL, where

PQVL D \�[email protected];t/�TK

hPET1PET2PRT 1

2P�

12 PP 1

iT; (19)

where @VL is the generalized gradient of VL and the entry 1 in (19) accommodates for the expressionof M as time-dependent in (15). Because VL .y; t/ is continuously differentiable,

PQVL � rVLKhPET1PET2PRT 1

2P�

12 PP 1

iT; (20)

where rVL ,hET1 ET2 RTM 2P

1212RT PM.t/R

i. After using the calculus for KŒ� from [30]

and substituting expressions from (5), (6), (10) and (14), (20) may be written as

PQVL �ET1 .E2 ��1E1/CE

T2 ..LB ˝ Im/R ��2E2/

CRT�� .LB ˝ Im/ .Ks C IFm/

�PE2 C�2E2

�� .LB ˝ Im/ ˇK ŒSgn .E2/C QN C LBNd

� .LB ˝ Im/E2 �1

2PMR

�C1

2RT PMR �

�PE2 C�2E2

�T.Nd � ˇK ŒSgn .E2// ;

(21)



where K ŒSgn .E2/k D 1 if E2k > 0, K ŒSgn .E2/k D �1 if E2k < 0, K ŒSgn .E2/k 2 Œ�1; 1 ifE2k D 0, and here the subscript k denotes the kth vector entry [30]. The set in (21) reduces to ascalar because the right-hand side is continuous a:e: because of the structure of the error signals;that is, the right-hand side is continuous except for the Lebesgue negligible set of time instancesin which RT .LB ˝ Im/ ˇK ŒSgn .E2/ � RT .LB ˝ Im/ ˇK ŒSgn .E2/ ¤ ¹0º§. After cancel-ing common terms, using the Raleigh–Ritz theorem and triangle inequality, recalling that LB ispositive-definite and symmetric, and using the bounding strategy in (13), the scalar value PVL can beupper-bounded a:e: as

PVLa:e:6 12kE1k

2 C1

2kE2k

2 � �min .�1/ kE1k2 � �min .�2/ kE2k

2 C kRk � .kZk/ kZk

�RT .LB ˝ Im/Ks .LB ˝ Im/R �RT .LB ˝ Im/

2R:

(22)

Using the definition of the auxiliary gain constant ˆ, which is positive given the sufficient gainconditions in (17) and the positive-definite property of LB (note that the product .LB ˝ Im/

2 ispositive-definite because LB is positive-definite and symmetric), (22) is rewritten as

PVLa:e:6 �ˆ kZk2 � �min ..LB ˝ Im/Ks .LB ˝ Im// kRk

2 C kRk � .kZk/ kZk ;

where the product .LB ˝ Im/Ks .LB ˝ Im/ is positive-definite becauseKs is positive-definite andLB is positive-definite and symmetric. After completing the squares, PVL is again upper-boundeda:e: by

PVLa:e:6 �

�ˆ �

�2 .kZk/

4�min ..LB ˝ Im/Ks .LB ˝ Im//

�kZk2 :

Provided the gains ks;i are selected such that the respective minimum eigenvalues are sufficientlylarge such that y.0/ 2 SD, there exists a constant c 2 R>0 such that

PVLa:e:6 �c kZk2 (23)

for all y 2 D. Thus, the inequalities in (16) and (23) show that VL 2 L1 and, therefore,E1; E2; R 2L1. A simple analysis of the closed-loop error system shows that the remaining signals are alsobounded. Furthermore, from (23), [32, Corollary 1] can be used to show c kZk2 ! 0 as t !1 forall y.0/ 2 SD. Because the vector Z contains the vector E1, kE1k ! 0 as t ! 1. By Lemma 1,d .qi ;Conv ¹ql j l 2 VLº/! 0 8i 2 VF , that is, each follower agent’s state converges to the convexhull spanned by the leaders’ states.

Note that the controller in (8) is distributed in communication; however, because the stabilizingset of initial conditions SD depends on the graph-dependent matrix LB , the gains ks;i must beselected in a centralized manner before execution of the control. However, the set SD can be madearbitrarily large to include any initial condition y.0/ by increasing the minimum eigenvalues of thegains ks;i to increase the minimum eigenvalue of the matrix .LB ˝ Im/Ks .LB ˝ Im/. �

6. SIMULATION

To demonstrate the robustness of the developed approach in performing containment control of fol-lower agents with respect to a set of leader agents, numerical simulations are performed for a groupof AUVs conducting surveillance. Each follower agent is modeled as a conventional, slender-bodied,

§Because of the construction of R in (6), the set of time instances ‚ ,®t 2 R>0 j RT .LB ˝ Im/ ˇK ŒSgn .E2/�

�RT .LB ˝ Im/ ˇK ŒSgn .E2/� ¤ ¹0º¯

can be represented by the union ‚ D [kD1;:::;Fm‚k , where ‚k ,®t 2 R>0 j E2k D 0^Rk ¤ 0

¯. Because the signal E2 W R>0 ! RFm is continuously differentiable, it can

be shown that ‚k is Lebesgue measure zero [28]. Because a finite union of Lebesgue measure zero sets is Lebesguemeasure zero,‚ is Lebesgue measure zero. Hence,‚ is Lebesgue negligible.



fully actuated AUV with nonlinear dynamics as described in [33] (see [26] for more information onAUV dynamics). The state of each AUV is composed of surge (x), sway (y), heave (´), roll ( ),pitch (�), and yaw ( ). Actuation of the AUV is modeled by three independent forces acting at thecenter of mass of the vehicle and three independent moments, which can be produced with a giventhruster configuration and an appropriate thruster mapping algorithm, such as that described in [34].

Four leader agents are used to direct five follower agents such that the follower agents’ statesconverge to the convex hull formed by the leaders’ time-varying states, which have initial positionsshown in Table I, identical initial velocities of Œ0:2m/s, 0m/s; 0:05m/s; 0 rad/s; 0 rad/s;�0:1 rad/sT ,and identical accelerations of Œ�0:02 sin .0:1t/ ;�0:02 cos .0:1t/ ; 0T m/s2 in surge, sway, andheave, respectively, such that the leaders form an inclined rectangle that translates in a helicaltrajectory. The follower AUVs have initial positions shown in Table II and identical initial veloc-ities of Œ2m/s; 0m/s; 0m/s; 0 rad/s; 0 rad/s; 0 rad/sT . All leader and follower agents have initialroll, pitch, and yaw of 0 rad. The network topology is shown in Figure 1, where aij D 1 if.j; i/ 2 EF [ EL and aij D 0 otherwise. As in [35], the external disturbances for the fol-lower AUVs are modeled as Œui sin .0:5t/ ; 0:5vi sin .0:25t/†; 0:2wi randT N in surge, sway, andheave, respectively, and 0 Nm in roll, pitch, and yaw, where ui ; vi ; wi 2 R represent the linearvelocities in surge, sway, and heave, respectively, of the i th follower agent and rand 2 RŒ�1;1�is a uniformly distributed random number generator. Identical gains for each follower agent i areselected as ks;i D diag .150; 150; 150; 1; 5; 70/, �i D diag .50; 50; 50; 0:1; 0:2; 0:2/, ˛1;i D 0:2,and ˛2;i D 0:1.

Table I. Leader initial positions insurge (x), sway (y), and heave (´).

Leader x (m) y (m) ´ (m)

1 �0:5 0.5 02 �0:5 �0:5 0

3 0.5 �0:5 1�

4 0.5 0.5 1�

Table II. Follower initial positions insurge (x), sway (y), and heave (´).

Follower x (m) y (m) ´ (m)

1 0 0.6 0.12 0.1 0.2 0.053 0.8 �0:2 04 �0:8 0.1 0.15 0.2 0.7 0.05

Figure 1. Network topology of leader (‘L’) and follower (‘F’) autonomous underwater vehicles.

†Correction added on 22 March 2016, after first online publication: “2.5” corrected to “0.25”.‡Correction added on 22 March 2016, after first online publication: “0” corrected to “1”.



Surge (m)-3 -2 -1 0 1 2 3

Sw

ay (

m)

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5time: 0 s

time: 15 s

time: 30 s

time: 45 s

AUV 1AUV 2AUV 3AUV 4AUV 5

Figure 2. [Top view] Follower autonomous underwater vehicle trajectories in the surge (x) and sway (y)dimensions. At the labeled time instances, the black outline represents the projection of the leaders’ convexhull onto surge and sway, black squares represent the follower positions, and circles within the leader outline

represent the equilibrium trajectories of the followers.

Surge (m)-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Hea

ve (

m)

0

0.5

1

1.5

2

2.5

3

3.5

time: 0 s

time: 15 s

time: 30 s

time: 45 s AUV 1AUV 2AUV 3AUV 4AUV 5

Figure 3. [Front view] Follower autonomous underwater vehicle trajectories in the surge (x) and heave (´)dimensions. At the labeled time instances, the black outline represents the projection of the leaders’ convexhull onto surge and heave, black squares represent the follower positions, and circles within the leader outline

represent the equilibrium trajectories of the followers.

The agents’ trajectories are shown in Figures 2 and 3. At the labeled time instances, the blackoutline represents the projection of the leaders’ convex hull onto the labeled dimensions, the blacksquares represent the follower agent positions, and the circles within the leader outline representthe equilibrium trajectories of the follower agents (which is a function of the network topologyand leader trajectories, i.e., when kE1k � 0). The containment errors, that is, each dimension ofthe error signal e1;i , are shown in Figure 4 for each agent, where ej1;i indicates the j th dimensionof the vector e1;i . In agreement with the analysis of the developed controller, Figures 2–4 indicatethat the follower agents cooperatively become contained within the leader convex hull and convergeto the containment equilibrium trajectories, despite the effects of model uncertainty and unknowntime-varying exogenous disturbances. The Euclidean norms of the overall AUV force and momentactuation, shown in Figure 5, demonstrate that reasonable actuation levels are used.



-5

0

5AUV 1 AUV 2 AUV 3 AUV 4 AUV 5

-2

0

2

-1

0

1

-0.5

0

0.5

-0.5

0

0.5

Time (s)0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

-0.2

0

0.2

Figure 4. Containment errors of the follower autonomous underwater vehicles in surge, sway, heave, roll,pitch, and yaw.

0 5 10 15 20 25 30 35 40 45

Ove

rall

forc

e no

rm (

N)

0

20

40

60

80

100


Time (s)0 5 10 15 20 25 30 35 40 45

Ove

rall

mom

ent n

orm

(N

m)

0

1

2

3

4

5


Figure 5. Euclidean norms of the follower autonomous underwater vehicle control efforts.



7. CONCLUSION

A distributed controller was developed for the cooperative containment control of autonomous net-worked follower agents with a set of network leaders, which is a generalization of the popular singleleader–follower network. The developed continuous controller provides robustness to input distur-bances and uncertain, nonlinear Euler–Lagrange dynamics such that the state of each follower agentasymptotically converges to the convex hull spanned by the leaders’ time-varying states for an arbi-trary number of leaders. Some notable assumptions are that the agents’ dynamics and disturbancesare smooth, the graph containing the follower agents is connected, and at least one follower agentis connected to at least one leader. Simulation results are provided to demonstrate the disturbancerejection capability of the developed controller. Future work may include extension of the givenstability analysis to show stability of the networked system when using the controller in (8) for ageneral digraph, such as in [18]; the limitation of the current analysis is that it requires the matrixLB to be positive-definite, which is not necessarily the case for a follower agent network topologythat is directed. Future research on cooperative containment control may also include the develop-ment of a distributed controller that is subject to the effects of practical communication, such ascommunication delays and packet dropouts (cf. [36–39]).

APPENDIX

Proof of Lemma 1Because the eigenvalues of LB are positive by Assumptions 5 and [9, Lemma 4.1], the diago-nal entries are positive, and the off-diagonal entries are nonpositive, all entries of the matrix L�1Bare nonnegative [40, Theorem 2.3]. Because of the structure of the Laplacian matrix L, we havethat L1FCL D 0FCL, which implies LB1F C LL1L D 0L, where 1 is a column vector ofones with the denoted dimension. Because LB is invertible, 1F D �L�1B LL1L, which impliesthat row sums of �L�1B LL add to one, where each entry of the matrix �L�1B LL is nonnegativebecause L�1B has only nonnegative entries and LL has only nonpositive entries. Thus, the prod-

uct ��L�1B LL

�˝ Im

�QL can be represented as

�qTd1; : : : ; qT

dF

�TD �

��L�1B LL

�˝ Im

�QL 2

RFm, where qdi 2 Rm such that qdi DPl2¹1;:::;Lº

��L�1B LL

�ilql 8i 2 ¹1; : : : ; F º, where

Œ�ij denotes the matrix entry of the i th row and j th column,Pl2¹1;:::;Lº

��L�1B LL

�ilD 1

8i 2 ¹1; : : : ; F º, and��L�1B LL

�il> 0 8i 2 ¹1; : : : ; F º ;8l 2 ¹1; : : : ; Lº by the aforemen-

tioned conclusions. Because the convex hull for the set S , ¹ql j l 2 VLº is defined as Conv¹Sº ,®Pl2VL ˛lql j .8l W R 3 ˛l > 0/ ^

Pl2VL ˛l D 1

¯[41], we have that qdi 2 Conv ¹ql j l 2 VLº

8i 2 ¹1; : : : ; F º; in other words, the vectors stacked in the product ��L�1B LL

�˝ Im

�QL are

within the convex hull formed with the leader agents’ states. Suppose that kE1k ! 0. Then.LB ˝ Im/�1E1 ! 0, which implies thatQF C

��L�1B LL

�˝ Im

�QL

! 0 by (4), and

thus, QF D�qTLC1; : : : ; q

TLCF

�T! �

��L�1B LL

�˝ Im

�QL D

�qTd1; : : : ; qT

dF

�T. Hence,

d .qi ;Conv ¹ql j l 2 VLº/! 0 8i 2 VF . �

ACKNOWLEDGEMENT

The authors would like to thank the AFRL Mathematical Modeling and Optimization Institute andNSF, contract/grant numbers 1161260 and 1217908, for the financial support.

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